1. ENE/EIE 325 Electromagnetic Fields and Waves
Lecture 12
Time-Varying Fields, Faraday’s Law, and Electromotive Force

2. Faraday’s Experiment
The question addressed is: If a current can generate a magnetic field, then can a magnetic field generate a current?
An experiment similar to that conducted to answer
that question is shown here. Two sets of windings
are placed on a shared iron core. In the lower set,
a current is generated by closing the switch as shown.
In the upper set, any induced current is registered by
the ammeter.

3. Effect of Increasing B
Closing the switch results in current that initially
increases with time in the lower windings. Magnetic
flux density B is generated, which now links the upper
windings; these are found to carry current, Iind .
This current exists as long as B increases with time.
Note the direction of the induced current and the
position of the ammeter.
More specifically, the induced current is proportional
to the time rate of change of the magnetic flux
(the surface integral of B over S ).

4. Steady State Once steady state is reached, in which B is at a constant
value, the induced current is found to be zero.

5. Effect of Decreasing B
Opening the switch results in current that initially
decreases with time in the lower windings.
Magnetic flux density B collapses, and the upper
windings carry current, Iind in the opposite
direction from before. This current exists as long
as the magnetic flux decreases with time.
Once B is reduced completely to zero,
the induced current is zero as well.

6. A Simpler Demonstration
Consider a single turn of wire, through which an
externally-applied magnetic flux is present. The flux
varies with time.
A current, Iind , is generated in the wire loop as
a result of the changing magnetic flux.

7. Replacing Current with Electric Field
Current in the wire is caused by electric field, E, that pushes the
charge around the wire. In this illustration, the wire is removed,
and the electric field remains. An integration contour, C, is also
shown, that coincides with E.

8. Electromotive Force (emf)
Next, a unit normal vector to the surface, n, is shown. The
relative directions of n, and the contour C are defined by the
right-hand convention: Right hand thumb in direction of n;
fingers then curve in the direction of C.
The electromotive force, or emf, is defined as the closed path
integral of E about C:

9. Electromagnetic induction in slowly varying field
Faradays’ law
“The electromotive force (emf) is produced by the time-varying magnetic field.”
Lenz’s law
“The induced emf produced in such a direction as to produce
a current whose flux, if added to the original flux, would reduce the magnitude of the emf.”

10. Electromagnetic induction in time-varying field
Electromagnetic induction is the phenomenon in which the current or the electromotive force (emf) is induced in a
conductor due to a change in the magnetic field near the conductor.
An electromotive force is merely a voltage (V) that arises from conductors moving in a magnetic field or from changing magnetic fields.

11. Faraday’s law Lenz’s law
The electromotive force (emf) is produced by the time-varying magnetic field.
The minus sign is an indication that the induced voltage acts to produce an
opposing flux (Lenz’s law).

12. Faraday’s Law of Induction
Faraday’s Law states that the induced emf around a closed path
is equal to the time rate of change of the magnetic flux through
the area surrounded by the path:
where the flux is:
Note the use of the normal unit vector here, which determines
the sign of the flux, and ultimately the emf.

13. Faraday’s Law in Detail
Using the definitions of emf and the magnetic flux in this situation:
becomes:
When we strictly apply this rule to the illustration at the left,
do we conclude that B is increasing or decreasing with time?
Question:

14. Alternative Expression for Faraday’s Law (Another Maxwell Equation)
First move the time derivative
into the interior of the right-hand
integral:
Second, use Stokes’ theorem to
write the line integral of E as the
surface integral of its curl:
Third, simplify by noting that both
integrals are taken over the same surface:
Finally, obtain the point form of
Faraday’s Law:

15. Example Start with a uniform (but time-varying) field: Then
As this calculation will work for any radius, we can replace the fixed
radius with a variable radius and write:

16. Another Way… This time use: where Therefore: Surviving z-directed
curl component with
radial variation only
Multiply through by and integrate both sides:
Obtain, as before:
An observation: The given B field does not satisfy all of Maxwell’s
equations! More about this later…

17. Sources of the time-varying magnetic field
Alternating magnetic field, H(t)
The steady magnetic field in a moving closed conducting path H H t
VS. t

18. Faraday’s law Observe the voltmeter needle when
move the magnet in and out
flip the magnet polarity
increase a number of coil turns
Click at the image to see the tutorial

19. Moving the magnet into the coil
Q: What is a direction of the induced current??
moving in
induced current

20. Moving the magnet out of the coil
Q: What is a direction of the induced current??
moving out
induced current

21. Flipping the magnet polarity, moving the magnet into the coil
Q: What is a direction of the induced current??
moving in
moving out
induced current

22. Flipping the magnet polarity, moving the magnet out of the coil
Q: What is a direction of the induced current??
moving out
induced current

23. Increasing the number of turn, moving the magnet into the coil
Q: What is a direction of the induced current??
moving in
moving out
induced current

24. Increasing the number of turn, moving the magnet out of the coil
Q: What is a direction of the induced current??
moving out
moving out
induced current

25. More general form of Faraday’s law
where N is the number of turns in a coil filamentary conductor.
 = flux linkage = N

26. Electromagnetic induction applications
Electrical transformer
Electrical motor
Generator

27. Electrical transformer
Click at the image to see how it works

28. How does an electrical motor work?
Click at the image to see how it works
The electrical motor converts
the electrical energy into the
mechanical energy.

29. How does an electrical generator work?
Click at the image to see how it works
The electrical generator converts
the mechanical energy into the
electrical energy.

30. Electrical motor vs. electrical generator
Click at the image to see how different they are

31. Induced emf from a Moving Closed Path
In the case shown here, the flux within a closed path is increasing with time as a result of the sliding bar, moving at constant velocity v.
The conducting path is assumed perfectly-conducting, so that the induced emf appears entirely across the voltmeter.
The magnetic flux enclosed by the conducting path is
Therefore:
where the y dimension varies with time.

32. Finding the Direction of E
..so we have:
Direction of integration
path of E through conductor
(by right hand convention)
The E integration path direction is determined by the
right hand convention (thumb of right hand in
direction of n,fingers in direction of closed path).
This is the direction of dL as shown in the figure.
From the equation, we see that is
negative, meaning that E will point from
terminals 2 to 1 as shown. As we have a
perfectly-conducing path, E will exist only
at the voltmeter.

33. Motional EMF
The sliding bar contains free charges (electrons) that experience
Lorentz forces as they move through the B field:
Recall:
and so..
This result is the motional field intensity:
..and so now the motional emf is the closed path integral of Em over the same path as before.
Note that the motional field intensity
Exists only in the part of the circuit
that is in motion.

34. Reprise of the Original Problem
We now have the motional emf:
..which in the current problem becomes:
or the same result as before!
as always, this path
integral is taken according
to the right hand convention,
leading to the ordering of limits shown here

35. Two Contributions to emf
For a moving conducting path in an otherwise uniform field, the emf is composed entirely of
motional emf:
When the B field is time-varying as well, then both effects need to be taken into account.
The total emf is:
Transformer emf
Motional emf
Together, the two effects are really just…

36. Ex2 A perfectly conducting filament containing a small 500Ω resistor is formed into a square, as illustrated below. Find I(t) if = (a) 0.3cos(120πt−30) Teslas; (b) 0.4cos[π(ct−y)] µT, where c = 3×108 m/s.

37. Ex3 Let mT, find
flux passing through the surface z = 0, 0 < x < 20 m,
and 0 < y < 3 m at t = 1 S.
b) value of closed line integral around the surface specified
above at t = 1 S.

38. Ex4 A moving conductor is located on the conducting rail as shown at time t = 0,
a) find emf when the conductor is at rest at x = 0.05 m and T.

39. b) find emf when the conductor is moving with the speed m/s.